Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe

Abstract

The way we organise perturbation theory is of fundamental importance both for computing the observables of relevance and for extracting fundamental physics out of them. If on one hand the different ways in which the perturbative observables can be written make manifest different features (e.g. symmetries as well as principles such as unitarity, causality and locality), on the other hand precisely demanding that some concrete features are manifest lead to different ways of organising perturbation theory. In the context of flat-space scattering amplitudes, a number of them are already known and exploited, while much less is known for cosmological observables. In the present work, we show how to systematically write down both the wavefunction of the universe and the flat-space scattering amplitudes, in such a way that they manifestly show physical poles only. We make use of the invariant definition of such observables in terms of cosmological polytopes and their scattering facet. In particular, we show that such representations correspond to triangulations of such objects through hyperplanes identified by the intersection of their facets outside of them. All possible triangulations of this type generate the different representations. This allows us to provide a general proof for the conjectured all-loop causal representation of scattering amplitudes. Importantly, all such representations can be viewed as making explicit a subset of compatible singularities, and our construction provides a way to extend Steinmann relations to higher codimension singularities for both the flat-space scattering amplitudes and the cosmological wavefunction.

Amplitudes meet Cosmology: A (Scalar) Primer

“…They allow to sharpen questions about the relation between cosmology and flat-space physics, providing a concrete framework that makes precise the sense in which Lorentz invariance as well as flat-space unitarity and causality emerge 111,112 . This led to novel, combinatorial, proofs of the flat-space cutting rules 111 as well as Steinmann relations 112 and the causal representation for scattering amplitudes 113 . The cosmological polytopes are objects living in projective space and are endowed with a differential form, called canonical form, with just logarithmic singularities on the boundary of the polytopes.…”

“…As the wavefunction universal integrand is associated to the canonical form, all the singularities of the former are encoded in the boundaries of the latter. This allowed to further understand the physics encoded in such singularities and formulate novel constraints on the wavefunction in terms of compatibility conditions on multiple singularities 112,113 . Furthermore, the cosmological polytope description allowed to realise a deeper relation of the universal integrand to flat-space scattering amplitudes.…”

“…G with different propagating states (i.e. different ν e 's) can be related via a recursive relation 109 (113) This relation can be proven considering the integrand in the definition of…”

“…G , applying the total-time translation operator −i v∈V ∂ ηv and integrating over the conformal times. As this integral is zero due to the Bunch-Davies condition as well to the vanishing of the bulk-to-bulk propagators at the boundary, acting with the total-time translation in the integrand provides an equation which, after suitable manipulation, returns (113). Importantly, the action of the total-time translation operator on the bulk-to-bulk propagators relates them to the ones for propagating states with order parameter ν e − 1.…”

“…(1) any graph contribution to the wavefunction coefficients with external arbitrary scalars can be obtained from the same graph contribution but with external conformally-coupled scalars via differential operators acting in energy space; (2) the graph contributions different propagating states characterised by order parameters which differ by an integer are related to each other via (113). For the case of heavy mass states for which the order parameter is purely imaginary, (113) relates them to states which do no appear to be in the Hilbert space. Hence, for heavy mass states it stays as a purely formal expression, while it becomes meaningful for light states, which are characterised by having ν e real.…”